The actual propositions quoted by Ptolemy as proved by Apollonius among others show mathematically at what points, under each of the two hypotheses, the apparent forward motion changes into apparent retrogradation and vice versa, or the planet appears to be stationary.

Moreover, the distance that the planet travels between one retrogradation and the next is quite variable.

From the sum of all these small motions will result a daily retrogradation of the points of intersection of the equator and the ecliptic  that is to say, of the equinoxes; and though it is extremely small, yet, as it is daily repeated in the same direction, the effect at the end of a year will be very sensible.